is an \(\mathbb {Z}_2\) -subspace of \((P_t)_n.\)  For later convenience, let us denote by \(\widetilde{\Phi _*}(C) := \widetilde{\Phi ^0}(C) \bigcup \widetilde{\Phi ^{>0}}(C).\)  Let \(\omega \)  be a weight vector of degree \(n\)  and let \(({C}^{\otimes t}_{n})^{\omega }:= {C}^{\otimes t}_{n}\cap (P_t)_{n}^{\omega }.\) In [1]}, Sum sets up the following conjecture, which plays a crucial role in studying the minimal set of \(\mathcal {A}_2\) -generators for \(P_t\)  in each positive degree.
